Form the differential equation of all circles which pass through … - Vidyadip

Question

Form the differential equation of all circles which pass through origin and whose centres lie on Y-axis.

✓

Answer

It is given that, circles pass through origin and their centres lie on Y-axis. Let (0, k) be the centre of the circle and radius is k.
So, the equation of circle is
$(\text{x}-0)^2+(\text{y}-\text{k})^2=\text{k}^2$
$\Rightarrow\text{x}^2+(\text{y}-\text{k})^2=\text{k}^2$
$\Rightarrow\text{x}^2+\text{y}^2-2\text{ky}=0$
$\Rightarrow\frac{\text{x}^2+\text{y}^2}{2\text{y}}=\text{k}\ ......(\text{i})$
On differentiating Eq. (i) w.r.t.x, we get
$\frac{2\text{y}\Big(2\text{x}+2\text{y}\frac{\text{dy}}{\text{dx}}\Big)-({\text{x}^2+\text{y}^2})\frac{2\text{dy}}{\text{dx}}}{4\text{y}^2}=0$
$\Rightarrow4\text{y}\Big(\text{x}+\text{y}\frac{\text{dy}}{\text{dx}}\Big)-2(\text{x}^2+\text{y}^2)\frac{\text{dy}}{\text{dx}}=0$
$\Big[4\text{y}^2-2(\text{x}^2+\text{y}^2)\Big]\frac{\text{dy}}{\text{dx}}+4\text{xy}=0$
$\Rightarrow(4\text{y}^2-2\text{x}^2-2\text{y}^2)\frac{\text{dy}}{\text{dx}}+4\text{xy}=0$
$\Rightarrow(2\text{y}^2-2\text{x}^2)\frac{\text{dy}}{\text{dx}}+4\text{xy}=0$
$\Rightarrow(\text{y}^2-\text{x}^2)\frac{\text{dy}}{\text{dx}}+2\text{xy}=0$
$\Rightarrow(\text{x}^2-\text{y}^2)\frac{\text{dy}}{\text{dx}}-2\text{xy}=0$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "4 Marks" from DIFFERENTIAL EQUATIONS→DIFFERENTIAL EQUATIONS — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

A factory manufactures two types of screws A and B, each type requiring the use of two machines, an automatic and a hand-operated. It takes 4 minutes on the automatic and 6 minutes on the hand-operated machines to manufacture a packet of screws ‘A’ while it takes 6 minutes on the automatic and 3 minutes on the hand-operated machine to manufacture a packet of screws ‘B’. Each machine is available for at most 4 hours on any day. The manufacturer can sell a packet of screws ‘A’ at a profit of 70 paise and screws ‘B’ at a profit of Rs. 1. Assuming that he can sell all the screws he manufactures, how many packets of each type should the factory owner produce in a day in order to maximize his profit? Formulate the above LPP and solve it graphically and find the maximum profit.

Find the angle of intersection of the curves y² = 4ax and x² = 4by.

Evaluate the following integrals:
$\int\limits^{\frac{\pi}{3}}_{\frac{\pi}{6}}\frac{\sqrt{\sin\text{x}}}{\sqrt{\sin\text{x}}+\sqrt{\cos\text{x}}}\text{ dx}$

Show that the points $\hat{\text{i}}-\hat{\text{j}}+3\hat{\text{k}}$ and $3\hat{\text{i}}+3\hat{\text{j}}+3\hat{\text{k}}$ are equidistant from the plane $\vec{\text{r}}\cdot(5\hat{\text{i}}+2\hat{\text{j}}-7\hat{\text{k}})+9=0$

Evaluate: $\int\frac{\sin(\text{x} - \text{a})}{\sin(\text{x + a})}\text{ dx}.$

Find the maximum and the minimum values, if any, without using derivaives of the following functions:

f(x) = x³ - 1 on R.

There are three coins. One is two-headed coin (having head on both faces), another is biased coin that comes up heads 75% of the times and third is also a biased coin that comes up tail 40% of the times. One of the three coins is chosen at random and tossed, and it shows heads. What is the probability that it was the two-headed coin?

Find the equation of the plane passing through the points whose coordinates are (-1, 1, 1) and (1, -1, 1) and perpendicular to the plane x + 2y + 2z = 5.

Solve the following differential equations:
$\text{xy}\frac{\text{dy}}{\text{dx}}=(\text{x}+2)(\text{y}+2),\text{y}(1)=-1$

Solve the following differential equation:
$\frac{\text{dy}}{\text{dx}} = \sec(\text{x}+\text{y})$