Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11

# If the Abscissae and Ordinates of Two Points P And Q Are Roots of the Equations X2 + 2ax − B2 = 0 And X2 + 2px − Q2 = 0 Respectively, Then Write the Equation of the Circle With Pq as Diameter. - Mathematics

If the abscissae and ordinates of two points P and Q are roots of the equations x2 + 2ax − b2 = 0 and x2 + 2px − q2 = 0 respectively, then write the equation of the circle with PQ as diameter.

#### Solution

The roots of the equations x2 + 2ax − b2 = 0 and x2 + 2px − q2 = 0 are $- a \pm \sqrt{a^2 + b^2}$  and $- p \pm \sqrt{p^2 + q^2}$ . Therefore, the coordinates of P and Q are $\left( - a + \sqrt{a^2 + b^2}, - p + \sqrt{p^2 + q^2} \right) \text { and } \left( - a - \sqrt{a^2 + b^2}, - p - \sqrt{p^2 + q^2} \right)$   -a+a2+b2, -p+p2+q2 and -a-a2+b2, -p-p2+q2 , respectively.
So, the required equation of the circle is

$\left( x + a - \sqrt{a^2 + b^2} \right)\left( x + a + \sqrt{a^2 + b^2} \right) + \left( y + p - \sqrt{p^2 + q^2} \right)\left( y + p + \sqrt{p^2 + q^2} \right) = 0$

$\Rightarrow \left( x + a \right)^2 - a^2 - b^2 + \left( y + p \right)^2 - p^2 - q^2 = 0$

$x^2 + y^2 + 2ax + 2yp - p^2 - q^2 = 0$

Concept: Circle - Standard Equation of a Circle
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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 24 The circle
Q 4 | Page 38