Advertisements
Advertisements
Question
The normal to the curve x2 = 4y passing through (1, 2) is _____________ .
Options
x + y = 3
x − y = 3
x + y = 1
x − y = 1
none of these
Advertisements
Solution
\[\text { Given }: \]
\[ x^2 = 4y\]
\[ \Rightarrow 2x = 4\frac{dy}{dx}\]
\[ \Rightarrow \frac{dy}{dx} = \frac{2x}{4} = \frac{x}{2}\]
\[\text { Slope of the tangent } = \left( \frac{dy}{dx} \right)_\left( 1, 2 \right) =\frac{1}{2}\]
\[\text { Slope of the normal,}m=\frac{- 1}{\text{ Slope of the tangent }}=\frac{- 1}{\frac{1}{2}}=-2\]
\[\text { Also }, \]
\[\left( x_1 , y_1 \right) = \left( 1, 2 \right)\]
\[ \therefore \text { Equation of the normal }\]
\[ = y - y_1 = m \left( x - x_1 \right)\]
\[ \Rightarrow y - 2 = - 2 \left( x - 1 \right)\]
\[ \Rightarrow y - 2 = - 2x + 2\]
\[ \Rightarrow 2x + y = 4\]
Notes
None of the given options is correct.
APPEARS IN
RELATED QUESTIONS
The equation of tangent at (2, 3) on the curve y2 = ax3 + b is y = 4x – 5. Find the values of a and b.
Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is `6sqrt3` r.
Find the equations of the tangent and normal to the curve `x^2/a^2−y^2/b^2=1` at the point `(sqrt2a,b)` .
Find points on the curve `x^2/9 + "y"^2/16 = 1` at which the tangent is parallel to x-axis.
Find the equation of the normal to curve y2 = 4x at the point (1, 2).
The slope of the tangent to the curve x = t2 + 3t – 8, y = 2t2 – 2t – 5 at the point (2,– 1) is
(A) `22/7`
(B) `6/7`
(C) `7/6`
(D) `(-6)/7`
At what points on the circle x2 + y2 − 2x − 4y + 1 = 0, the tangent is parallel to x-axis?
Find the points on the curve x2 + y2 = 13, the tangent at each one of which is parallel to the line 2x + 3y = 7 ?
Find the points on the curve y = x3 where the slope of the tangent is equal to the x-coordinate of the point ?
Find the equation of the tangent and the normal to the following curve at the indicated point y = x4 − bx3 + 13x2 − 10x + 5 at (0, 5) ?
Find the equation of the tangent and the normal to the following curve at the indicated point 4x2 + 9y2 = 36 at (3cosθ, 2sinθ) ?
Find the equation of the tangent and the normal to the following curve at the indicated point \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text { at } \left( \sqrt{2}a, b \right)\] ?
Find the equation of the tangent and the normal to the following curve at the indicated points x = θ + sinθ, y = 1 + cosθ at θ = \[\frac{\pi}{2}\] ?
Find the equation of a normal to the curve y = x loge x which is parallel to the line 2x − 2y + 3 = 0 ?
Find the equation of the tangent to the curve x2 + 3y − 3 = 0, which is parallel to the line y= 4x − 5 ?
Find the equation of the tangents to the curve 3x2 – y2 = 8, which passes through the point (4/3, 0) ?
Find the condition for the following set of curve to intersect orthogonally \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text { and } xy = c^2\] ?
Show that the curves \[\frac{x^2}{a^2 + \lambda_1} + \frac{y^2}{b^2 + \lambda_1} = 1 \text { and } \frac{x^2}{a^2 + \lambda_2} + \frac{y^2}{b^2 + \lambda_2} = 1\] intersect at right angles ?
Find the slope of the normal at the point 't' on the curve \[x = \frac{1}{t}, y = t\] ?
The point on the curve y2 = x where tangent makes 45° angle with x-axis is ______________ .
The point on the curve y2 = x where tangent makes 45° angle with x-axis is ____________________ .
The angle between the curves y2 = x and x2 = y at (1, 1) is ______________ .
If the curve ay + x2 = 7 and x3 = y cut orthogonally at (1, 1), then a is equal to _____________ .
The equation of the normal to the curve x = a cos3 θ, y = a sin3 θ at the point θ = π/4 is __________ .
The slope of the tangent to the curve x = t2 + 3t − 8, y = 2t2 − 2t − 5 at the point (2, −1) is _____________ .
The normal at the point (1, 1) on the curve 2y + x2 = 3 is _____________ .
Find the equation of the tangent line to the curve `"y" = sqrt(5"x" -3) -5`, which is parallel to the line `4"x" - 2"y" + 5 = 0`.
Find the equation of all the tangents to the curve y = cos(x + y), –2π ≤ x ≤ 2π, that are parallel to the line x + 2y = 0.
The tangent to the curve given by x = et . cost, y = et . sint at t = `pi/4` makes with x-axis an angle ______.
At what points on the curve x2 + y2 – 2x – 4y + 1 = 0, the tangents are parallel to the y-axis?
The curve y = `x^(1/5)` has at (0, 0) ______.
For which value of m is the line y = mx + 1 a tangent to the curve y2 = 4x?
The number of common tangents to the circles x2 + y2 – 4x – 6x – 12 = 0 and x2 + y2 + 6x + 18y + 26 = 0 is
The Slope of the normal to the curve `y = 2x^2 + 3 sin x` at `x` = 0 is
If the curves y2 = 6x, 9x2 + by2 = 16, cut each other at right angles then the value of b is ______.
The number of values of c such that the straight line 3x + 4y = c touches the curve `x^4/2` = x + y is ______.
If the tangent to the conic, y – 6 = x2 at (2, 10) touches the circle, x2 + y2 + 8x – 2y = k (for some fixed k) at a point (α, β); then (α, β) is ______.
