Let $\mathrm{P}\left(\mathrm{a}_{1}, \mathrm{~b}_{1}\right)$ and $\mathrm{Q}\left(\mathrm{a}_{2}, \mathrm{~b}_{2}\right)$ be two distinct points on a circle with center $\mathrm{C}(\sqrt{2}, \sqrt{3})$. Let O be the origin and OC be perpendicular to both CP and CQ . If the area of the triangle OCP is $\frac{\sqrt{35}}{2}$, then $\mathrm{a}_{1}^{2}+\mathrm{a}_{2}^{2}+\mathrm{b}_{1}^{2}+\mathrm{b}_{2}^{2}$ is equal to \_\_\_\_ .

JEE Main · 2023 · 30 Jan 2023 · Shift 2 · Circle

Let

P (a_{1}, b_{1})

and

Q (a_{2}, b_{2})

be two distinct points on a circle with center

C (2, 3)

. Let O be the origin and OC be perpendicular to both CP and CQ . If the area of the triangle OCP is

\frac{35}{2}

, then

a_{1}^{2} + a_{2}^{2} + b_{1}^{2} + b_{2}^{2}

is equal to ____ .

Answer: 24

Solution

With

C = (2, 3)

O C = 2 + 3 = 5

, and

O C ⊥ C P

makes triangle

O C P

right-angled at

C

. Its area

\frac{1}{2} \cdot O C \cdot C P = \frac{35}{2}

gives

C P = 7

, which is the circle's radius, so

C Q = 7

too. By the right angles at

C

O P^{2} = O C^{2} + C P^{2} = 5 + 7 = 12

and

O Q^{2} = O C^{2} + C Q^{2} = 12

. Hence

a_{1}^{2} + b_{1}^{2} + a_{2}^{2} + b_{2}^{2} = O P^{2} + O Q^{2} = 24

JEE Main · 2023 · 30 Jan 2023 · Shift 2 · Integer answer

Practise more: all JEE Main PYQs on Circle →

More JEE Main questions on Circle

Let PQ and MN be two straight lines touching the circle $x^{2} + y^{2} - 4 x - 6 y - 3 = 0$ at the points A and B respectively. Let O be the centre of the circle and $∠ AOB = π /3$ . Then the locus of the point of intersection of the lines PQ and MN is:
JEE Main 2026View solution →
If P is a point on the circle $x^{2} + y^{2} = 4$ , Q is a point on the straight line $5 x + y + 2 = 0$ and $x - y + 1 = 0$ is the perpendicular bisector of PQ, then 13 times the sum of abscissa of all such point P is ______.
JEE Main 2026View solution →
Let the set of all values of r, for which the circles $(x + 1)^{2} + (y + 4)^{2} = r^{2}$ and $x^{2} + y^{2} - 4 x - 2 y - 4 = 0$ intersect at two distinct points be the interval ( $α$ , $β$ ). Then $α β$ is equal to
JEE Main 2026View solution →
Let a circle of radius 4 pass through the origin O , the points $A (- 3 a, 0)$ and $B (0, - 2 b)$ , where a and b are real parameters and $ab \neq = 0$ . Then the locus of the centroid of $△ OAB$ is a cir …
JEE Main 2026View solution →
Let $y = x$ be the equation of a chord of the circle $C_{1}$ (in the closed half-plane $x \geq 0$ ) of diameter 10 passing through the origin. Let $C_{2}$ be another circle described on the given chord as its diameter. If the equation of the chord …
JEE Main 2026View solution →
Let the circle $x^{2} + y^{2} = 4$ intersect $x$ -axis at the points $A (a, 0), a > 0$ and $B (b, 0)$ . Let $P (2 cos α, 2 sin α), 0 < α < \frac{π}{2}$ and $Q (2 cos β, 2 sin β)$ be two points …
JEE Main 2026View solution →