Advertisements
Advertisements
Question
Find a point on the curve y = (x − 2)2 at which the tangent is parallel to the chord joining the points (2, 0) and (4, 4).
Advertisements
Solution
If a tangent is parallel to the chord joining the points (2, 0) and (4, 4), then the slope of the tangent = the slope of the chord.
The slope of the chord is `(4 - 0)/(4 - 2) = 4/2 = 2`
Now, the slope of the tangents to the given curve at a point (x, y) is given by,
`dy/dx = 2(x - 2)`
Since the slope of the tangent = slope of the chord, we have:
2(x – 2) = 2
`\implies` x – 2 = 1
`\implies` x = 3
When x = 3, y = (3 – 2)2 = 1
Hence, the required point is (3, 1).
APPEARS IN
RELATED QUESTIONS
Find the slope of the tangent to the curve y = (x -1)/(x - 2), x != 2 at x = 10.
Find the slope of the normal to the curve x = acos3θ, y = asin3θ at `theta = pi/4`
Find the points on the curve y = x3 at which the slope of the tangent is equal to the y-coordinate of the point.
For the curve y = 4x3 − 2x5, find all the points at which the tangents passes through the origin.
Prove that the curves x = y2 and xy = k cut at right angles if 8k2 = 1. [Hint: Two curves intersect at right angle if the tangents to the curves at the point of intersection are perpendicular to each other.]
Find the equation of the tangent to the curve `y = sqrt(3x-2)` which is parallel to the line 4x − 2y + 5 = 0.
Find the slope of the tangent and the normal to the following curve at the indicted point x = a cos3 θ, y = a sin3 θ at θ = π/4 ?
Find the slope of the tangent and the normal to the following curve at the indicted point y = (sin 2x + cot x + 2)2 at x = π/2 ?
Find the points on the curve \[\frac{x^2}{9} + \frac{y^2}{16} = 1\] at which the tangent is parallel to y-axis ?
Find the equation of the tangent and the normal to the following curve at the indicated point y = x2 at (0, 0) ?
Find the equation of the tangent and the normal to the following curve at the indicated point y = x2 + 4x + 1 at x = 3 ?
Find the equation of the tangent and the normal to the following curve at the indicated point \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \text { at } \left( x_1 , y_1 \right)\] ?
Find the equation of the tangent and the normal to the following curve at the indicated point y2 = 4x at (1, 2) ?
Find the equations of all lines of slope zero and that are tangent to the curve \[y = \frac{1}{x^2 - 2x + 3}\] ?
Find the equation of the tangent to the curve x2 + 3y − 3 = 0, which is parallel to the line y= 4x − 5 ?
Find the angle of intersection of the following curve 2y2 = x3 and y2 = 32x ?
Find the angle of intersection of the following curve x2 + 4y2 = 8 and x2 − 2y2 = 2 ?
Prove that the curves xy = 4 and x2 + y2 = 8 touch each other ?
Find the condition for the following set of curve to intersect orthogonally \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text { and } xy = c^2\] ?
Write the angle made by the tangent to the curve x = et cos t, y = et sin t at \[t = \frac{\pi}{4}\] with the x-axis ?
Write the angle between the curves y = e−x and y = ex at their point of intersections ?
Write the equation of the tangent drawn to the curve \[y = \sin x\] at the point (0,0) ?
The angle between the curves y2 = x and x2 = y at (1, 1) is ______________ .
The equation of the normal to the curve 3x2 − y2 = 8 which is parallel to x + 3y = 8 is ____________ .
If the curves y = 2 ex and y = ae−x intersect orthogonally, then a = _____________ .
The point on the curve y = 6x − x2 at which the tangent to the curve is inclined at π/4 to the line x + y= 0 is __________ .
The normal at the point (1, 1) on the curve 2y + x2 = 3 is _____________ .
The normal to the curve x2 = 4y passing through (1, 2) is _____________ .
The tangent to the curve given by x = et . cost, y = et . sint at t = `pi/4` makes with x-axis an angle ______.
Find the co-ordinates of the point on the curve `sqrt(x) + sqrt(y)` = 4 at which tangent is equally inclined to the axes
Find the angle of intersection of the curves y = 4 – x2 and y = x2.
The tangent to the curve y = e2x at the point (0, 1) meets x-axis at ______.
The points on the curve `"x"^2/9 + "y"^2/16` = 1 at which the tangents are parallel to the y-axis are:
Find the equation of the tangent line to the curve y = x2 − 2x + 7 which is parallel to the line 2x − y + 9 = 0.
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 tangentto the curve `x= t^2 + 3t - 8, y = 2t^2 - 2t - 5` at the point `(2, -1)` is
If the tangent to the curve y = x + siny at a point (a, b) is parallel to the line joining `(0, 3/2)` and `(1/2, 2)`, then ______.
